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1.4c Inverse Relations and Inverse Functions

1.4c Inverse Relations and Inverse Functions. Homework: p. 129 39-61 odd. Definition: Inverse Relation. The ordered pair ( a , b ) is in a relation if and only if the ordered pair ( b , a ) is in the inverse relation. Consider the “Do Now”:. Which of these relations are functions???.

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1.4c Inverse Relations and Inverse Functions

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  1. 1.4c Inverse Relations and Inverse Functions Homework: p. 129 39-61 odd

  2. Definition: Inverse Relation The ordered pair (a, b) is in a relation if and only if the ordered pair (b, a) is in the inverse relation. Consider the “Do Now”: Which of these relations are functions??? These relations are inverses of each other! (the x- and y-values are simply switched!)

  3. The HLT!!! The Horizontal Line Test The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original function in at most one point. Fails the HLT miserably!!! So, its inverse is not a function…

  4. Practice Problems Is the graph of each relation a function? Does the relation have an inverse that is a function? Not a function A function Has an inverse that is a function Has an inverse that is a function

  5. Practice Problems Is the graph of each relation a function? Does the relation have an inverse that is a function? A function Not a function Has an inverse that is not a function Has an inverse that is not a function

  6. More Definitions A relation that passes both the VLT and HLT is called one-to-one. (since every x is paired with a unique y and every y is paired with a unique x…) If f is a one-to-one function with domain D and range R, then the inverse function of f, denoted f , is the function with domain R and range D defined by –1 if and only if

  7. Finding an Inverse Algebraically 1. Determine that there is an inverse function by checking that the original function is one-to-one. Note any restrictions on the domain of the function. 2. Switch x and y in the formula of the original function. 3. Solve for y to obtain the inverse function. State any restrictions of the domain of the inverse.

  8. Finding an Inverse Algebraically Find the inverse of the given function algebraically: Check the graph  is the function one-to-one?

  9. Finding an Inverse Graphically The Inverse Reflection Principle The points (a, b) and (b, a) in the coordinate plane are symmetric with respect to the line y = x. The points (a, b) and (b, a) are reflections of each other across the line y = x.

  10. Finding an Inverse Graphically The graph of a function is shown. Is the function one-to-one? Sketch a graph of the inverse of the function. Yes!!! y = x

  11. And one more new tool:The Inverse Composition Rule A function f is one-to-one with inverse function g if and only if f (g(x)) = x for every x in the domain of g, and g(f (x)) = x for every x in the domain of f We can use this rule to algebraically verify that two functions are inverses… observe…

  12. More Practice Show algebraically that the given functions are inverses.

  13. More Practice Show that the given function has an inverse and find a rule for that inverse. State any restrictions of the domains of the function and its inverse. Check the graph  Is f one-to-one? where where Let’s graph the inverse together with the original function… where where

  14. Whiteboard Problems… Find a formula for . Give the domain of , including any restrictions “inherited” from f.

  15. Whiteboard problems… Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

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